Rosh Hodesh means "head of the month". If the month that we are going from has 30 days in it then the 30th and the first of the next month will be celebrated as Rosh Hodesh, if the month has 29 days, then the celebration is on the first and is only one day.

During Temple times, before the calendar was set, people would look to the night sky to see the moon. When they saw just a sliver of the moon they went to the Sanhedrim (the religious court) and told them about what they saw. When at least two witnesses were confirmed as actually seeing the moon then it was declared a new month. 

The day after, there would be celebrations announced by blowing the Shofar. People did not go to work. many went to Jerusalem where they sacrificed a special new-month offering. Afterwards they would gather for a family feast. 

Women were given the day off from their labors. This was due to their refusing to give their jewelry when Aaron made the golden idol.

Today Rosh Hodesh is celebrated with special prayers and customs. Usually the new moon blessing is done following the Havdallah service on Saturday night following the appearance. It is wonderful to do this outside in the open air so you can see the moon. The Shabbat before Rosh Hodesh is called Shabbat Mevarekhim (Shabbat of Blessing) this is also where the Yehi Ratzon (Prayer for the new month) is said.